Theorem un0.1 45134

Description: ⊤ is the constant true, a tautology (see df-tru 1545). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
un0.1.1	⊢ (   ⊤   ▶   𝜑   )
un0.1.2	⊢ (   𝜓   ▶   𝜒   )
un0.1.3	⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )

Assertion

Ref	Expression
un0.1	⊢ (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1

Step	Hyp	Ref	Expression
1		un0.1.1	. . . 4 ⊢ (   ⊤   ▶   𝜑   )
2	1	in1 44927	. . 3 ⊢ (⊤ → 𝜑)
3		un0.1.2	. . . 4 ⊢ (   𝜓   ▶   𝜒   )
4	3	in1 44927	. . 3 ⊢ (𝜓 → 𝜒)
5		un0.1.3	. . . 4 ⊢ (   (   ⊤   ,   𝜓   )   ▶   𝜃   )
6	5	dfvd2ani 44939	. . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃)
7	2, 4, 6	uun0.1 45133	. 2 ⊢ (𝜓 → 𝜃)
8	7	dfvd1ir 44929	1 ⊢ (   𝜓   ▶   𝜃   )

Syntax hints:  ⊤wtru 1543  (   wvd1 44925  (   wvhc2 44936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-vd1 44926  df-vhc2 44937

This theorem is referenced by:  sspwimpVD  45274